Generalized complex geometry of pure backgrounds in 10 and 11 dimensions

Pure backgrounds are a natural generalization of supersymmetric Calabi–Yau compactifications in the presence of flux. They are described in the language of generalized $SU\left(d\right)\times SU\left(d\right)$ structures and generalized complex geometry, and they exhibit some interesting general patterns: the internal manifold is generalized Calabi–Yau, whereas the Ramond–Ramond flux is exact in a precise sense, as discussed in this paper. We have shown that although these two characteristics do persist in the case of generic 10-dimensional Euclidean type II pure backgrounds, they do not capture the full content of supersymmetry. We also discuss the uplift of real Euclidean type IIA pure backgrounds to supersymmetric backgrounds of Lorentzian 11-dimensional supergravity. (paper)